Please share your thoughts!

We ended up discussing something quite similar at the recent MIRI workshop, which I attended! We decided that the Gaifman condition proposed in that paper is probably a bad idea. I proposed something along these lines as a potential modification of my logical prior, and Will Sawin showed that it leads to some bad results.

I may post a draft of the theorem here soon. For now, note that the unrestricted Gaifman condition is not satisfied by any computably approximable probability distribution. If it were, we could condition on the true atomic statements of number theory and then converge to the correct probability (1 or 0) for all statements of number theory: we would converge to the correct answer (1 or 0) for statements with a single quantifier, because existential quantifiers are proved from examples and universals get the right probability by Gaifman. But then this is like conditioning on the true statements with one quantifier; so the statements with two quantifiers then get the right probability by the same argument. And so, all statements get the right probability, by induction on the number of quantifiers. Hence Gaifman distributions get all the statements of number theory right.

This obviously is not computably approximable, making the Gaifman condition seem useless for probability distributions which capture uncertainty about mathematics.

Will proved that even much-restricted versions of the Gaifman condition will lead to bad results. I hope to post more details soon.

Best,

Abram

I have two comments on this statement:

(1) I do not regard the Gaifman&Cournot prior (G&C) to model uncertainty about true but unprovable statements, but to model induction.
If you condition on the Peano axioms, G&C indeed assigns probability 1 to all statements true in the standard model of natural numbers, including e.g. Goedel's sentence.
But if you're unsure about whether e.g. 1+1=2, you should regard Peano axioms as universal hypotheses, provide arithmetic relations as "data", and then the posterior will converge to 1.

(2) As with Solomonoff, G&C is not meant to be a computable theory but a theoretical gold standard to approximate and aim at.
There is one difference, Solomonoff is limit-computable, while G&C is even outside the arithmetic hierarchy, see [GS82],
i.e. G&C is much "worse" when viewed from a computability point of view,
but on the other hand it is inductively more powerful.

Of course, ultimately we need to approximate both, and which properties survive to which degree depends on the approximation.
But at least for "hypotheses of practical interest" we need to preserve the Gaifman condition.
I do not see how or know of any other method that gets this right.

Cheers,

Marcus
 

I initially intended to reply more quickly, but was uncertain of whether a draft of Will Sawin's result would be ready for wider distribution. (We've distributed it a bit, but have been unsure of whether the result was new.)